Distributed Semi‐Supervised Single‐Index Model With Corruption

ABSTRACT Modern dataset is often large in scale, necessitating the development of distributed training methods. In this paper, we focus on distributed learning for the single‐index model (SIM). Particularly, we consider the presence of unlabelled covariate information and assume that an fraction of labels may be arbitrarily corrupted. To address the challenges of distributed training, we first propose a modified rank regression loss function that eliminates local bias in distributed training. To leverage the unlabelled covariates, we develop two distributed semi‐supervised algorithms tailored for low‐dimensional and high‐dimensional settings, respectively. We theoretically demonstrate that the inclusion of unlabelled data accelerates distributed training. Notably, our method is inherently robust to a moderate fraction of label corruptions, regardless of how the corrupted labels are distributed across the worker machines. Simulation studies are provided to validate the effectiveness of our approach.